Lesson 5: Comparing Linear, Exponential, and Quadratic Models — Practice Questions

1. 1. A data set includes the points $(0, 5)$, $(1, 15)$, and $(2, 45)$. If the data represents an exponential function, what is the constant ratio between consecutive y-values? ___

2. 2. Which property indicates that a data set, with evenly spaced x-values, can be modeled by an exponential function?

   • A. The difference between consecutive y-values is constant.
   • B. The ratio of consecutive y-values is constant.
   • C. The second differences of the y-values are constant.
   • D. The y-values are always increasing.

3. 3. Which exponential function models the data set containing the points $(1, 50)$, $(2, 25)$, and $(3, 12.5)$?

   • A. $y = 50(0.5)^x$
   • B. $y = 100(0.5)^x$
   • C. $y = 50(2)^x$
   • D. $y = 100(2)^x$

4. 4. The following points model exponential decay: $(0, 80)$, $(1, 20)$, and $(2, 5)$. What is the constant decay factor for this function? ___

5. 5. An exponential function $y = ab^x$ passes through the points $(2, 18)$ and $(3, 54)$. What is the initial value, $a$? ___

6. 6. Which of the following data sets has consecutive x-values with a constant difference, making it suitable for difference analysis?

   • A. (-2, 4), (0, 9), (2, 14), (4, 19)
   • B. (0, 5), (1, 7), (3, 11), (4, 13)
   • C. (1, 10), (4, 8), (5, 7), (8, 4)
   • D. (-5, 2), (-4, 3), (-2, 5), (0, 7)

7. 7. A set of data points is given by $(0, 12), (4, 19), (8, 28), (12, 39)$. What is the constant difference, $\Delta x$, between consecutive x-values? $\Delta x = $ ___.

8. 8. Why is it essential to confirm that consecutive x-values have a constant difference before using first or second differences to identify a function type?

   • A. It guarantees the function is linear.
   • B. It ensures that patterns in the y-value differences are reliable indicators of the function model.
   • C. It is only necessary for data sets with negative numbers.
   • D. It confirms that the y-values are increasing at a constant rate.

9. 9. The following data points have x-values that are evenly spaced: $(3, 8), (7, 11), (k, 14), (15, 17)$. What is the value of $k$? $k = $ ___.

10. 10. Consider the data set $(1, 20), (2, 18), (4, 14), (7, 8)$. Is this data set appropriate for analysis using the method of finite differences?

    • A. Yes, the x-values have a constant difference.
    • B. No, the y-values are decreasing.
    • C. No, the x-values do not have a constant difference.
    • D. Yes, all the data points lie on a straight line.